On the Fujita exponent for a Hardy-H\'{e}non equation with a spatial-temporal forcing term

TL;DR

This paper investigates the existence, uniqueness, and blow-up of solutions for a higher-order parabolic Hardy-Hénon equation with spatial-temporal forcing, identifying a Fujita critical exponent depending on the forcing term's growth rate.

Contribution

It establishes conditions for global existence and blow-up, and determines the Fujita critical exponent as a function of the forcing term's temporal growth rate.

Findings

Global solutions exist for small initial data when parameters satisfy certain conditions.

Solutions blow up under specific growth conditions of the forcing term and initial data.

The Fujita critical exponent varies with the temporal growth rate of the forcing term.

Abstract

The purpose of this work is to analyze the wellposedness and the blow-up of solutions of the higher-order parabolic semilinear equation \[ u_t+(-\Delta)^{d}u=|x|^{\alpha}|u|^{p}+\zeta(t){\mathbf w}(x) \ \quad\mbox{for } (x,t)\in\mathbb{R}^{N}\times(0,\infty), \] where $d\in (0,1)\cup \mathbb{N}$, $p>1$, $-\alpha\in(0,\min(2d,N))$ or $\alpha\geq 0$ and $\zeta$ as well as ${\mathbf w}$ are suitable given functions. Given $p\geq \frac{N-2d\sigma+\alpha}{N-2d\sigma-2d}$ and setting $p_c=\frac{N(p-1)}{2d+\alpha}$, $\ell=\frac{N p_c}{N+2(\sigma+1)d p_c}$, we prove that for any data $u_0\in L^{p_c,\infty}(\mathbb{R}^N)$ and $\textbf{w}\in L^{\ell,\infty}(\mathbb{R}^N)$ with small norms there exists a unique global-in-time solution under the hypotheses $\zeta(t)=t^{\sigma}$, $\sigma\in (-1,0)$ and $N>2d$ in the space $C_{b}([0,\infty);L^{p_c,\infty}(\mathbb{R}^N))$. As a by-product, small Lebesgue data global existence follows and in particular, unconditional uniqueness holds in $C_{b}([0,\infty);L^{p_c}(\mathbb{R}^N))$ provided $p\in (\frac{N+\alpha}{N-2d},\infty)$. If either $m\in (-\infty,0]$ and $p\in (1,\frac{N-2dm+\alpha}{N-2dm-2d})$ or $m>0$ and $p>1$ where $\zeta(t)=O(t^m)$, $t\rightarrow\infty$ ($m\in \mathbb{R}$), then all solutions blow up under the additional condition $\int_{\mathbb{R}^N}\textbf{w}(x)\,dx>0$. As a consequence, we deduce that the corresponding Fujita critical exponent is a function of $\sigma$ and reads $p_{F}(\sigma)=\frac{N-2d\sigma+\alpha}{N-2d\sigma-2d}$ if $-1<\sigma<0$ and infinity otherwise.